fixed is_nthpow_residue

[abhinav-anand-addepar]

Fixes #18394

• ntheory
  • is_nthpow_residue no longer raises ValueError when a < 0
  • polynomial_congruence recognizes x**n + a = 0 mod m as a special case

[sympy-bot]

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• ntheory

  • is_nthpow_residue no longer raises ValueError when a < 0 (#18398 by @abhinav28071999)

  • polynomial_congruence recognizes x**n + a = 0 mod m as a special case (#18398 by @abhinav28071999)

This will be added to https://github.com/sympy/sympy/wiki/Release-Notes-for-1.6.

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Fixes #18394 

<!-- BEGIN RELEASE NOTES -->
* ntheory
  * `is_nthpow_residue` no longer raises ValueError when a < 0
  * `polynomial_congruence` recognizes x**n + a = 0 mod m as a special case
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

[codecov]

Codecov Report

Merging #18398 into master will increase coverage by 0.011%.
The diff coverage is 87.5%.

@@              Coverage Diff              @@
##            master    #18398       +/-   ##
=============================================
+ Coverage   75.292%   75.304%   +0.011%     
=============================================
  Files          633       637        +4     
  Lines       166944    167071      +127     
  Branches     39396     39417       +21     
=============================================
+ Hits        125697    125812      +115     
- Misses       35709     35719       +10     
- Partials      5538      5540        +2

[smichr]

Suggested change

	if coefficients[0] == 1 and all(v == 0 for v in coefficients[1:-1]):
	if coefficients[0] == 1 and 1 + coefficients[-1] = sum(coefficients):

[abhinav-anand-addepar]

When the coefficient is negative this might give wrong result. I think I should take mod m of all coefficients then this suggestion can be correctly implemented.

[smichr]

No, what you have is better.

[smichr]

But why not

if not any(coefficients[1: -1]):
    g = igcd(coefficients[0], coefficients[-1])
    if coefficients[0]//g == 1:
        return nthroot_mod(-coefficients[-1]//g, rank - 1, m, True)

In that case 2*x**2 + 8 will be reduced to x**2 + 4.

[abhinav-anand-addepar]

@smichr yes but in this case also we have to take mod m of the coefficients. For example:
for x**5 + 16 x + 7 = 0 mod 8. But this can be reduced to x**5 + 7. But (if not any(coefficients[1:-1])) will not catch this unless coefficients = coefficients mod m.
Also by dividing the coefficients[0] with gcd might result in loss of some roots.
For example : 2*x**2 + 8 = 0 mod 10has roots [1,4,6,9] but x**2 + 4 = 0 mod 10 has
roots [4,6] because mod_inv(2,10) does not exist.

[oscarbenjamin]

I don't know whether the changes here are good but the release note can be improved. Try to write a release note that makes sense for end users. They don't care about internal details: they just want know what has changed from the outside. You can see examples of release notes here:
https://github.com/sympy/sympy/wiki/Release-Notes-for-1.5

[smichr]

I edited release notes.